Gravitational potential is a scalar quantity that represents the energy associated with a unit mass in a gravitational field. The gravitational field is a vector field that describes the spatial variations in potential. In the context of the Earth, local variations in the gravity field are caused, for example, by deviations in the surface of the Earth from a geometric sphere, surface and subsurface geology, water tides, atmospheric tides, and the change in the relative position of the earth, moon, and the sun.
A gravitational field can be decomposed into three mutually-orthogonal components Γx, Γy, Γz shown in FIG. 1. For reference, a coordinate system is used in which the X-axis corresponds to the north-south alignment, the Y-axis corresponds to the east-west alignment, and the Z-axis corresponds to the up-down alignment. The z vector (Γz) will have the largest magnitude and the x, y vectors (Γx, Γy) will have respective magnitudes that are a function of the location of the observation point relative to any mass inhomogeneities.
Each of the three vector components Γx, Γy, Γz of the gravity field has a gradient parallel to each of the three mutually-perpendicular coordinate axes. The gradient describes the rate of change of a vector as a function of movement in the three orthogonal directions. For example, the gradient
      ∂          Γ      x            ∂    y  describes how the Γx vector changes with movement in the y direction.
The full gravity gradient tensor, Γij, is a matrix of nine components or gradients formed by the three orthogonal components of gravity force (Γx, Γy, Γz) as they change along three orthogonal spatial directions (x, y, z), as defined below:
                                                        ∂                              Γ                x                                                    ∂              x                                ⁢                                          ⁢                                    ∂                              Γ                x                                                    ∂              y                                ⁢                                          ⁢                                    ∂                              Γ                x                                                    ∂              z                                      ⁢                                  ⁢                                            ∂                              Γ                y                                                    ∂              x                                ⁢                                          ⁢                                    ∂                              Γ                y                                                    ∂              y                                ⁢                                          ⁢                                    ∂                              Γ                y                                                    ∂              z                                      ⁢                                  ⁢                                            ∂                              Γ                z                                                    ∂              x                                ⁢                                          ⁢                                    ∂                              Γ                z                                                    ∂              y                                ⁢                                          ⁢                                    ∂                              Γ                z                                                    ∂              z                                                          [        1        ]            
Gravity is a conservative field, and some of the fundamental properties of it are that:
                                                        ∂                              Γ                x                                                    ∂              y                                =                                    ∂                              Γ                y                                                    ∂              x                                      ,                                            ∂                              Γ                x                                                    ∂              z                                =                                    ∂                              Γ                z                                                    ∂              x                                      ,                                                            ∂                Γ                            ⁢                                                          ⁢              y                                      ∂              z                                =                                    ∂                              Γ                z                                                    ∂              y                                                          [        2        ]                                                                    ∂                              Γ                x                                                    ∂              x                                +                                                    ∂                Γ                            ⁢                                                          ⁢              y                                      ∂              y                                +                                    ∂                              Γ                z                                                    ∂              z                                      =        constant                            [        3        ]            As a consequence of expressions [2] and [3], only five of the nine tensor elements shown at [1] are independent.
The measurement of the spatial variations (gradients) of the gravity field is known as “gravity gradiometry.” The gravity gradient is a gradient of acceleration and so the appropriate units are acceleration units divided by distance units; that is, (meter/sec2)/meter. Typical gravity gradients (e.g., as measured during exploration, etc.) are exceedingly small, and are usually measured as nanoseconds squared, where: 1 ns−2=10−9 s−2=1 Eötvös unit (Eo).
The device that is used to measure these gradients, such as occurs when traversing the Earth's terrain (see, FIG. 3, terrain 36) or moving past a massive object, is known as a “gravity gradiometer.” In a typical gravity gradiometer, plural opposed pairs of accelerometers are moved at a constant velocity along an orbital path about a spin axis. Information from each accelerometer, which is in the form of a sinusoidally-varying analog signal, provides information as to the lateral acceleration, including the gravity field, sensed by the accelerometers as they orbit the spin axis.
For a gradiometer having its spin axis aligned along the field lines in an ideally uniform and unperturbed gravity field, each accelerometer experiences the same acceleration forces as it proceeds along its orbital path. But in situations in which the local gravity field is perturbed by the presence of one or more masses and/or the spin axis is tilted relative to the local vertical field lines, each accelerometer will experience different accelerations throughout its orbit. The quantitative output of each accelerometer, coupled with its rotary position, provides information related to the local gravity gradients.
Typically (but not necessarily), a gravity gradiometer will use either four accelerometers or eight accelerometers. FIG. 2 depicts disc assembly 24 of a four-accelerometer gradiometer. The sensitive axes of accelerometers 38A, 38B, 38C, and 38D are positioned at 90° intervals in the plane of disc or rotor 30, as shown in FIG. 2. The “sensitive axis” of an accelerometer is the directional axis along which acceleration is detected. Ideally, accelerations applied to the accelerometer orthogonal to the sensitive axis are not detected. In order to measure the gravity force, the accelerometers shown in FIG. 2 point in the direction of measured force rather than acceleration. If the instrument is subject to an actual acceleration, the perceived force will be in the direction opposite to the acceleration.
With perfect alignment and with perfectly balanced scale-factors, the sum of any single pair of diametrically-opposed accelerometers cancels common linear accelerations, but is sensitive to the difference in gravity between the accelerometers and also to angular accelerations about an axis that is normal to the sensitive plane. When static, the sum of signals from such an accelerometer pair provides a single gravity gradient plus a measure of angular acceleration.
In existing eight-accelerometer gravity gradiometers, all accelerometers are evenly spaced around the circumference of a disc and processed as if they were two superimposed four-accelerometer gravity gradiometers, offset by 45°. With reference to FIG. 2, four additional accelerometers 38E, 38F, 38G, and 38H are shown in phantom at the appropriate locations to provide an eight-accelerometer gravity gradiometer.
Any single disc of either four or eight accelerometers measures only two gradients. (The four additional accelerometers in an eight-accelerometer design enable, through redundant measurements, an increase in the signal-to-noise ratio of the gravitational-field measurement. But they do not enable the measurement of additional gradients.)
To measure all independent elements of the gradient tensor on a moving vehicle, three discs rotating about three mutually-orthogonal axes are used. Typically, each disc carries a set of four accelerometers. Full tensor gravity gradiometer 10 is depicted in FIG. 3.
As depicted in FIG. 3, gravity gradiometer 10 has three disc assemblies 24, 26, and 28 having respective discs 30, 32, and 34. These discs are mounted in a respective plane that is coincident or parallel with one of the three body-axis planes such that the spin axis of the disc is either coincident with or parallel to the body axis that is normal to the mounting plane. Furthermore, each disc is characterized by orthogonal disc axes that lie in but rotate relative to the mounting plane.
For example, disc 30 lies in the X-Y body-axis plane, has a spin axis Zs that is parallel to the Z body axis (i.e., the x-y coordinates of Zs are x=C1 and y=C2 where C1 and C2 are constants), and includes orthogonal disc axes XD and YD. As disc 30 rotates (in a counterclockwise direction in this example), disc axes XD and YD rotate relative to the non-rotating X and Y body axes. At the instant of time shown in FIG. 3, the XD and YD disc axes of disc 30 are respectively parallel and coincident with the X and Y body axes.
Disc 32 lies in a plane that is parallel to the Y-Z body-axis plane and has a spin axis Xs that is parallel to the X body axis. At the instant of time shown in FIG. 3, YD and ZD disc axes of disc 32 are respectively parallel and coincident with the Y and Z body axes. For clarity, little detail regarding disc 34 is provided. It can be seen, however, that disc 34 lies in a plane that is parallel to the Z-X body-axis plane and has a spin axis that is parallel to the Y body axis.
Disc assemblies 24, 26, and 28 each include two pairs of accelerometers. (Although gradiometer 10 includes two pairs of accelerometers per disc, measurements could be obtained with only one pair of accelerometers per disc.) For brevity, only disc assembly 24 will be discussed.
Disc assembly 24 includes respective pairs of accelerometers 38a, 38b and 38c, 38d. Each accelerometer 38a, 38b and 38c, 38d includes a respective input axis 40a, 40b and 40c, 40d along which the accelerometer measures a respective acceleration magnitude Aa, Ab, Ac, and Ad. Each accelerometer is mounted to disc 30 such that its input axis is a radius R from the spin axis Zs and is perpendicular to R. In some other embodiments, input axes 40a, 40b and 40c, 40d are oriented at other angles relative to R. This can occur intentionally or as a result of manufacturing imperfections.
Briefly, the gradient of the gravity field is measured as the difference in readings between the opposing pairs of accelerometers on each disc. Measurements from each disc assembly 24, 26, 28 can be resolved into two gradients in the plane of the respective rotating disc. The tensor components measured in the external coordinate axis directions are obtained by forming the appropriate linear combinations of the six outputs (i.e., two gradients from each disc).
It is understood that there are other elements of a gradiometer, such as a gimbaling system and processor and/or memory that are not shown in FIG. 3. More particularly, a full tensor gravity gradiometer includes a gimbaling system for rotational isolation. The gimbaling system typically includes a rotational sensor assembly, such as a gyroscope, for measuring rotational activity about the X, Y, and Z axes. Control signals derived from the sensor measurements are fed back to motors that are attached to the gimbal axes to reduce the rotations experienced by the gradiometer.
It is impossible, however, to completely eliminate such rotations, and tensor measurements will be additively corrupted by the presence of gradient signals due to these rotations. These additional, non-gravitational gradients are simple deterministic functions of the rotational rates. Therefore, the measurements from the gradiometer have these corrupting signals subtracted by the processor. In some embodiments, the processor is external to the gradiometer. In such embodiments, the gradiometer often includes a memory for storing measurement data for download to the external processor. In some further embodiments, the gradiometer includes a transmitter for transmitting measurement data to an external processor and/or memory.
The operation of the disc assemblies and the processing of signals therefrom to determine the gravity gradient are well understood to those skilled in the art. Further description of the operation of the disc assemblies and signal processing is provided, for example, in U.S. Pat. No. 6,799,459, which is incorporated herein by reference. See also, U.S. Pats. No. 5,357,802, 6,212,952, and 6,658,935, all incorporated herein by reference, which provide disclosure concerning four- and/or eight-accelerometer gravity gradiometers.
The gravity gradiometer described above is capable of extraordinary accuracy (i.e., accurate to about 1 Eo). Yet, it is subject to a variety of error mechanisms that can adversely affect its performance. A few such error mechanisms are described below.
Accelerometer Misalignment
The net misalignment, in the axial and radial directions, of the accelerometers on a disc can severely impact system performance. Although mechanical alignment of the accelerometer sensitive axes is carefully adjusted prior to commissioning the gradiometer, misalignments routinely remain. In certain gradiometer systems, the pendulous axes of the accelerometer coincide with the rotation axis. Therefore, by changing the accelerometer null position, fine realignment of the accelerometer in the axial direction is possible. But tangential realignment is not possible. In some other gradiometer systems, the accelerometers are mounted such that the pendulous axes have tangential as well as axial components. As a consequence, it is possible to realign the accelerometers electronically so that tangential misalignments are reduced. But doing so affects axial alignment. Unfortunately, axial and tangential alignments are not independently adjustable.
Instrument Rotation
Rotating gravity gradiometers intentionally modulate the gravity gradient via instrument rotation so that low-frequency gravity signals are moved away from the larger instrument error frequencies. The instrument errors, though no longer having power spectral density peaks coinciding with the modulated gravity gradient signals, still corrupt measurements at harmonics of the rotation rate and nearby frequencies. The strength of the instrument errors near harmonics of the rotation rate and near DC reduced the accuracy of the measurements at corresponding frequencies. By way of example, DC errors in the instrument frame cause the measurements of gradients near twice the rotation rate to be unusable.
Accelerometer Bias Matching and Scale Factor Balancing
Part of the difficulty in manufacturing gravity gradiometers comes from the need to balance, to a high degree, signal components of the various accelerometers composing the gradiometer measurement.
Examples of this include accelerometer (1) bias matching and (2) scale factor balance. Regarding (1), when building a gradiometer having a standard complement of four accelerometers, a substantially greater number of accelerometers must be made so that four can be selected that are sufficiently matched in bias values to enable the sum of the accelerometers to have a low net value. This low value is required to avoid instrumentation saturation problems and to keep noise problems related to offsetting a large sum at insignificant levels.
Regarding (2), gradiometers can operate in a maneuvering vehicle because the accelerometer pairs which are differenced in order to obtain a measurement of the gradient each sense the same maneuvering accelerations. In order for this to happen to the high degree of accuracy required, the accelerometers used in gradiometers have been required to have an alterable scale factor. This is required because it is practically impossible to ensure that, during manufacture, accelerometer scale factors will match sufficiently to accurately reject ambient accelerations when differenced.
Bias matching and scale factor balancing therefore add substantial complexity to the design and manufacture of gradiometer-ready accelerometers.
Unknown Rotational Rates
All systems that measure gravity gradients can be corrupted by unknown rotational rates. The usual solution for this has been to reduce rotational rates to a low level and, if necessary, apply compensations to the gradient measurements based on measurements of the residual rotational rates experienced by the gradiometer. In non-moving-base applications, this reduction of rotational rates is usually implemented in a passive mechanical fashion. For moving-base applications, and for instances in which ambient vibration levels would defeat a passive isolation system, active stabilization must be implemented. This is usually accomplished by placing all gradient sensors (i.e., the discs) on a common, fully-gimbaled platform on which a three-axis rotational stabilization system has been implemented.
The fully-isolated gimbal system adds substantial expense, size, and complexity to the gradiometer.
Design Constraints for Rejecting Rotational Accelerations
A measurement of an element of the gravity gradient tensor shows how one component of the gravity acceleration vector differs from the same vector component at a different location at the same time. Acceleration measurements at the two locations are differenced to remove the effect of common linear accelerations. But the measurements are not immune to differential motion. If both acceleration measurements are referenced to a common instrumentation block, the differential motion is caused by a rotation of the block around an axis that is orthogonal to both the position vector that separates the measurement locations and the measurement axis.
In the prior art, the rotational acceleration is rejected by adding a second acceleration difference measurement with sensitive axis and displacement vector interchanged from the first pair. This second pair of accelerometers also provides a measurement of the same gradient as the first pair, but in addition measures the rotational acceleration with opposite polarity. Adding this second measurement pair provides, as its primary purpose, the rejection of rotational accelerations.
Based on the aforementioned considerations, prior-art gradiometer designs are typically constrained with regard to the placement of accelerometers on the disc, the periodic nature of the disc rotation, and the demodulation frequency, among other parameters. In practice, these idealized configurations are never achieved. That is, the accelerometers in a complement are never exactly at 90°, the complements are never precisely regular in terms of position or orientation, the summation of never exactly reflects the true sum of the experienced accelerations, and the rotation is never truly periodic.
The various error mechanism discussed above (i.e., accelerometer misalignment, instrument rotation, scale and bias factor discrepancies, unknown rotational rates, and differential motion) are defined for reference herein and in the appended claims as “gravity gradiometer error mechanism(s).”
Identifying the Full Gravity Gradient Tensor
To identify the full gravity tensor, a minimum of five independent gradient components must be measured. Gradiometers that measure the full tensor typically use three accelerometer complements on three separate discs, all mounted on a stabilized platform. The drawbacks of this arrangement are the size of the resulting device, the relative vibration between sensors, and the spatial separation of the sensors. These latter two issues are additional error sources.
When stabilization sensors are separate from gradient sensors, relative vibration and resonance problems occur. When compensation sensors are separate from gradient sensors, problems related to relative motion, bandwidth, and dynamic response arise. Small gradiometers use modulation to measure two gradients, but with limited bandwidth. Large gradiometers have a wide bandwidth, but have been too large to use in a full tensor system. Current gradiometers are therefore limited to some degree by the number of gradients measured, bandwidth, size, vibration sensitivity, or problems of sensor distribution. Simply put, few if any gradiometer can measure the full tensor in a compact device.
In view of the foregoing, improvements in gravity gradiometer design would be beneficial.